In many real-world planning applications, e.g. dynamic design of experiments, autonomous driving and robot manipulation, it is necessary to evaluate candidate movement paths with respect to a safety cost function. Here, the continuous candidate paths need to be discretized first and, subsequently, evaluated onthe discretization points. The resulting quality of planned paths, thus, highly depends on the definition of the safety cost functions, and the resolution of the discretization. In this paper, we propose an approach for evaluating continuous candidate paths by employing an adaptive discretization scheme, with a probabilistic cost function learned from observations. The obtained path is then guaranteed to be epsilon-safe, i.e. the remaining risk of still finding an unsafe point on the trajectory is smaller than epsilon. The proposed approach is investigated theoretically, as well as empirically validated on several robotic path planning scenarios.

State-space models (SSMs) are a highly expressive model class for learning patterns in time series data and for system identification. Deterministic versions of SSMs (e.g., LSTMs) proved extremely successful in modeling complex time series data. Fully probabilistic SSMs, however, are often found hard to train, even for smaller problems. We propose a novel model formulation and a scalable training algorithm based on doubly stochastic variational inference and Gaussian processes. This combination allows efficient incorporation of latent state temporal correlations, which we found to be key to robust training. The effectiveness of the proposed PR-SSM is evaluated on a set of real-world benchmark datasets in comparison to state-of-the-art probabilistic model learning methods. Scalability and robustness are demonstrated on a high dimensional problem.

Learning time-series models is useful for many applications, such as simulation and forecasting. In this study, we consider the problem of actively learning time-series models while taking given safety constraints into account. For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure. The proposed approach generates data appropriate for time series model learning, i.e. input and output trajectories, by dynamically exploring the input space. The approach parametrizes the input trajectory as consecutive trajectory sections, which are determined stepwise given safety requirements and past observations. We analyze the proposed algorithm and evaluate it empirically on a technical application. The results show the effectiveness of our approach in a realistic technical use case.

PID control architectures are widely used in industrial applications. Despite their low number of open parameters, tuning multiple, coupled PID controllers can become tedious in practice. In this paper, we extend PILCO, a model-based policy search framework, to automatically tune multivariate PID controllers purely based on data observed on an otherwise unknown system. The system's state is extended appropriately to frame the PID policy as a static state feedback policy. This renders PID tuning possible as the solution of a finite horizon optimal control problem without further a priori knowledge. The framework is applied to the task of balancing an inverted pendulum on a seven degree-of-freedom robotic arm, thereby demonstrating its capabilities of fast and data-efficient policy learning, even on complex real world problems.

Learning control has become an appealing alternative to the derivation of control laws based on classic control theory. However, a major shortcoming of learning control is the lack of performance guarantees which prevents its application in many real-world scenarios. As a step towards widespread deployment of learning control, we provide stability analysis tools for controllers acting on dynamics represented by Gaussian processes (GPs). We consider differentiable Markovian control policies and system dynamics given as (i) the mean of a GP, and (ii) the full GP distribution. For both cases, we analyze finite and infinite time horizons. Furthermore, we study the effect of disturbances on the stability results. Empirical evaluations on simulated benchmark problems support our theoretical results.

Learning control has become an appealing alternative to the derivation of control laws based on classic control theory. However, a major shortcoming of learning control is the lack of performance guarantees which prevents its application in many real-world scenarios. As a step in this direction, we provide a stability analysis tool for controllers acting on dynamics represented by Gaussian processes (GPs). We consider arbitrary Markovian control policies and system dynamics given as (i) the mean of a GP, and (ii) the full GP distribution. For the first case, our tool finds a state space region, where the closed-loop system is provably stable. In the second case, it is well known that infinite horizon stability guarantees cannot exist. Instead, our tool analyzes finite time stability. Empirical evaluations on simulated benchmark problems support our theoretical results.

Sparse Gaussian process (GP) models provide an efficient way to perform regression on large data sets. The key idea is to select a representative subset of the available training data, which induces the sparse GP model approximation. In the past, a variety of selection criteria for GP approximation have been proposed, but they either lack accuracy or suffer from high computational costs. In this paper, we introduce a novel and straightforward criterion for successive selection of training points used for GP model approximation. The proposed algorithm allows a fast and efficient selection of training points, while being competitive in learning performance. As evaluation, we employ our approach in learning inverse dynamics models for robot control using very large data sets (e.g. 500.000 samples). It is demonstrated in experiments that our approximated GP model is sufficiently fast for real-time prediction in robot control. Comparisons with other state-of-the-art approximation techniques show that our proposed approach is significantly faster, while being competitive to generalization accuracy.

In many complex robot applications, such as grasping and manipulation, it is difficult to program desired task solutions beforehand, as robots are within an uncertain and dynamic environment. In such cases, learning tasks from experience can be a useful alternative. To obtain a sound learning and generalization performance, machine learning, especially, reinforcement learning, usually requires sufficient data. However, in cases where only little data is available for learning, due to system constraints and practical issues, reinforcement learning can act suboptimally. In this paper, we investigate how model-based reinforcement learning, in particular the probabilistic inference for learning control method (Pilco), can be tailored to cope with the case of sparse data to speed up learning. The basic idea is to include further prior knowledge into the learning process. As Pilco is built on the probabilistic Gaussian processes framework, additional system knowledge can be incorporated by defining appropriate prior distributions, e.g. a linear mean Gaussian prior. The resulting Pilco formulation remains in closed form and analytically tractable. The proposed approach is evaluated in simulation as well as on a physical robot, the Festo Robotino XT. For the robot evaluation, we employ the approach for learning an object pick-up task. The results show that by including prior knowledge, policy learning can be sped up in presence of sparse data.

For service robotics, localization is an essential component required in many applications, e.g. indoor robot navigation. Today, accurate localization relies mostly on high-end devices, such as A.R.T. DTrack, VICON systems or laser scanners. These systems are often expensive and, thus, require substantial investments. In this paper, our focus is on the development of a localization method using low-priced devices, such as cameras, while being sufficiently accurate in tracking performance. Vision data contains much information and potentially yields high tracking accuracy. However, due to high computational requirements vision-based localization can only be performed at a low frequency. In order to speed up the visual localization and increase accuracy, we combine vision information with robots odometry using a Kalman-Filter. The resulting approach enables sufficiently accurate tracking performance (errors in the range of few cm) at a frequency of about 35Hz. To evaluate the proposed method, we compare our tracking performance with the high precision A.R.T. DTrack localization as ground truth. The evaluations on real robot show that our low-priced localization approach is competitive for indoor robot localization tasks.

Task-space control of redundant robot systems based on analytical models is known to be susceptive to modeling errors. Data-driven model learning methods may present an interesting alternative approach. However, learning models for task-space tracking control from sampled data is an ill-posed problem. In particular, the same input data point can yield many different output values, which can form a nonconvex solution space. Because the problem is ill-posed, models cannot be learned from such data using common regression methods. While learning of task-space control mappings is globally ill-posed, it has been shown in recent work that it is locally a well-defined problem. In this paper, we use this insight to formulate a local kernel-based learning approach for online model learning for task-space tracking control. We propose a parametrization for the local model, which makes an application in task-space tracking control of redundant robots possible. The model parametrization further allows us to apply the kernel-trick and, therefore, enables a formulation within the kernel learning framework. In our evaluations, we show the ability of the method for online model learning for task-space tracking control of redundant robots.

Many real-world tasks require fast planning of highly dynamic movements for their execution in real-time. The success often hinges on quickly finding one of the few plans that can achieve the task at all. A further challenge is to quickly find a plan which optimizes a desired cost. In this paper, we will discuss this problem in the context of catching small flying targets efficiently. This can be formulated as a non-linear optimization problem where the desired trajectory is encoded by an adequate parametric representation. The optimizer generates an energy-optimal trajectory by efficiently using the robot kinematic redundancy while taking into account maximal joint motion, collision avoidance and local minima. To enable the resulting method to work in real-time, examples of the global planner are generalized using nearest neighbour approaches, Support Vector Machines and Gaussian process regression, which are compared in this context. Evaluations indicate that the presented method is highly efficient in complex tasks such as ball-catching.

Learning inverse kinematics of robots with redundant degrees of freedom (DoF) is a difficult problem in robot learning. The difficulty lies in the non-uniqueness of the inverse kinematics function. Existing methods tackle non-uniqueness by segmenting the configuration space and building a global solution from local experts. The usage of local experts implies the definition of an oracle, which governs the global consistency of the local models; the definition of this oracle is difficult. We propose an algorithm suitable to learn the inverse kinematics function in a single global model despite its multivalued nature. Inverse kinematics is approximated from examples using structured output learning methods. Unlike most of the existing methods, which estimate inverse kinematics on velocity level, we address the learning of the direct function on position level. This problem is a significantly harder. To support the proposed method, we conducted real world experiments on a tracking control task and tested our algorithms on these models.

Task-space tracking control is essential for robot manipulation. In practice, task-space control of redundant robot systems is known to be susceptive to modeling errors. Here, data driven learning methods may present an interesting alternative approach. However, learning models for task-space tracking control from sampled data is an ill-posed problem. In particular, the same input data point can yield many different output values which can form a non-convex solution space. Because the problem is ill-posed, models cannot be learned from such data using common regression methods. While learning of task-space control mappings is globally ill-posed, it has been shown in recent work that it is locally a well-defined problem. In this paper, we use this insight to formulate a local kernel-based learning approach for online model learning for task-space tracking control. For evaluations, we show in simulation the ability of the method for online model learning for task-space tracking control of redundant robots.

In recent years, learning models from data has become an increasingly interesting tool for robotics, as it allows straightforward and accurate model approximation. However, in most robot learning approaches, the model is learned from scratch disregarding all prior knowledge about the system. For many complex robot systems, available prior knowledge from advanced physics-based modeling techniques can entail valuable information for model learning that may result in faster learning speed, higher accuracy and better generalization. In this paper, we investigate how parametric physical models (e.g., obtained from rigid body dynamics) can be used to improve the learning performance, and, especially, how semiparametric regression methods can be applied in this context. We present two possible semiparametric regression approaches, where the knowledge of the physical model can either become part of the mean function or of the kernel in a nonparametric Gaussian process regression. We compare the learning performance of these methods first on sampled data and, subsequently, apply the obtained inverse dynamics models in tracking control on a real Barrett WAM. The results show that the semiparametric models learned with rigid body dynamics as prior outperform the standard rigid body dynamics models on real data while generalizing better for unknown parts of the state space.

The increasing complexity of modern robots makes it prohibitively hard to accurately model such systems as required by many applications. In such cases, machine learning methods offer a promising alternative for approximating such models using measured data. To date, high computational demands have largely restricted machine learning techniques to mostly offline applications. However, making the robots adaptive to changes in the dynamics and to cope with unexplored areas of the state space requires online learning. In this paper, we propose an approximation of the support vector regression (SVR) by sparsification based on the linear independency of training data. As a result, we obtain a method which is applicable in real-time online learning. It exhibits competitive learning accuracy when compared with standard regression techniques, such as v-SVR, Gaussian process regression (GPR) and locally weighted projection regression (LWPR).

Learning inverse kinematics has long been fascinating the robot learning community. While humans acquire this transformation to complicated tool spaces with ease, it is not a straightforward application for supervised learning algorithms due to non-convex learning problem. However, the key insight that the problem can be considered convex in small local regions allows the application of locally linear learning methods. Nevertheless, the local solution of the problem depends on the data distribution which can result into inconsistent global solutions with large model discontinuities. While this problem can be treated in various ways in offline learning, it poses a serious problem for online learning. Previous approaches to the real-time learning of inverse kinematics avoid this problem using smart data generation, such as the learner biasses its own solution. Such biassed solutions can result into premature convergence, and from the resulting solution it is often hard to understand what has been learned in that local region. This paper improves and solves this problem by presenting a learning algorithm which can deal with this inconsistency through re-weighting the data online. Furthermore, we show that our algorithms work not only in simulation, but we present real-time learning results on a physical Mitsubishi PA-10 robot arm.

High performance and compliant robot control requires accurate dynamics models which cannot be obtained analytically for sufficiently complex robot systems. In such cases, machine learning offers a promising alternative for approximating the robot dynamics using measured data. This approach offers a natural framework to incorporate unknown nonlinearities as well as to continually adapt online for changes in the robot dynamics. However, the most accurate regression methods, e.g. Gaussian processes regression (GPR) and support vector regression (SVR), suffer from exceptional high computational complexity which prevents their usage for large numbers of samples or online learning to date. Inspired by locally linear regression techniques, we propose an approximation to the standard GPR using local Gaussian processes models inspired by. Due to reduced computational cost, local Gaussian processes (LGP) can be applied for larger sample-sizes and online learning. Comparisons with other nonparametric regressions, e.g. standard GPR, nu-SVR and locally weighted projection regression (LWPR), show that LGP has higher accuracy than LWPR and close to the performance of standard GPR and nu-SVR while being sufficiently fast for online learning.

Learning in real-time applications, e.g., online approximation of the inverse dynamics model for model-based robot control, requires fast online regression techniques. Inspired by local learning, we propose a method to speed up standard Gaussian Process regression (GPR) with local GP models (LGP). The training data is partitioned in local regions, for each an individual GP model is trained. The prediction for a query point is performed by weighted estimation using nearby local models. Unlike other GP approximations, such as mixtures of experts, we use a distance based measure for partitioning of the data and weighted prediction. The proposed method achieves online learning and prediction in real-time. Comparisons with other nonparametric regression methods show that LGP has higher accuracy than LWPR and close to the performance of standard GPR and nu-SVR.